1. What is the equation of the circle with center (-4,-3) that passes through the point (6,2)?
A) (x-4)^2+(y-3)^2=25
B) (x-4)^2+(y-3)^2=125
C) (x-(-4))^2+(y-(-3))^2=25
D) (x-(-4))^2+(y-(-3))^2=25
I'm really not sure how to do this at all, so any help would be absolutely fantastic. Thank you so much.
I know it's one of the first two, but their centers are both positive (4,3) and I'm confused.
Does that mean it's not one of the first two? Because when I graphed C and D, they didn't match the answer choices.
To find the equation of a circle with a given center and a point that it passes through, you can use the general form of the equation for a circle:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
In this case, the center of the circle is (-4, -3), so h = -4 and k = -3. The point that the circle passes through is (6, 2).
To find the radius, you can use the distance formula between the center of the circle and the point on the circle:
r = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((6 - (-4))^2 + (2 - (-3))^2)
= sqrt(10^2 + 5^2)
= sqrt(100 + 25)
= sqrt(125)
= 5 * sqrt(5)
Now that you have the center (-4, -3) and the radius 5 * sqrt(5), you can substitute these values into the general equation of a circle:
(x - (-4))^2 + (y - (-3))^2 = (5 * sqrt(5))^2
(x + 4)^2 + (y + 3)^2 = 125
Comparing this equation with the given answer choices, we can see that option B) matches the equation:
(x - 4)^2 + (y - 3)^2 = 125
So, the equation of the circle with center (-4, -3) that passes through the point (6, 2) is B) (x - 4)^2 + (y - 3)^2 = 125.